Unifying Known Lower Bounds via Geometric Complexity Theory

Abstract

We show that most algebraic circuit lower bounds and relations between lower bounds naturally fit into the representation-theoretic framework suggested by geometric complexity theory (GCT), including: the partial derivatives technique (Nisan-Wigderson), the results of Razborov and Smolensky on AC^0[p], multilinear formula and circuit size lower bounds (Raz et al.), the degree bound (Strassen, Baur-Strassen), the connected components technique (Ben-Or, Steele-Yao), depth 3 algebraic circuit lower bounds over finite fields (Grigoriev-Karpinski), lower bounds on permanent versus determinant (Mignon-Ressayre, Landsberg-Manivel-Ressayre), lower bounds on matrix multiplication (Burgisser-Ikenmeyer, Landsberg-Ottaviani) (these last two were already known to fit into GCT), the chasms at depth 3 and 4 (Gupta-Kamath-Kayal-Saptharishi, Agrawal-Vinay, Koiran), matrix rigidity (Valiant) and others. That is, the original proofs, with what is often just a little extra work, already provide representation-theoretic obstructions in the sense of GCT for their respective lower bounds. This enables us to expose a new viewpoint on GCT, whereby it is a natural unification of known results and broad generalization of known techniques. It also shows that the framework of GCT is at least as powerful as previous methods, and gives many new proofs-of-concept that GCT can indeed provide significant asymptotic lower bounds. This new viewpoint also opens up the possibility of fruitful two-way interactions between previous results and the new methods of GCT, we provide several concrete suggestions of such interactions. For example, the representation-theoretic viewpoint of GCT naturally provides new properties to consider in the search for new lower bounds.